The Causal Action in Minkowski Space and Surface Layer Integrals

The Lagrangian of the causal action principle is computed in Minkowski space for Dirac wave functions interacting with classical electromagnetism and linearized gravity in the limiting case when the ultraviolet cutoff is removed. Various surface layer integrals are computed in this limiting case.

The theory of causal fermion systems is a recent approach to fundamental physics (see [10,5] and the references therein). In this approach, the physical equations are formulated via a variational principle, the causal action principle. In order to rewrite the resulting dynamics in a familiar way in terms of Dirac particles interacting with classical gauge fields, one considers the so-called continuum limit (see again [5] and the references therein). In the analysis of the continuum limit, one studies the Euler-Lagrange (EL) equations corresponding to the causal action principle in the limit ε ց 0 when an ultraviolet regularization is removed. In the present paper, we proceed differently and study instead the Lagrangian itself asymptotically as ε ց 0. This serves two different aims: First, we thus obtain an "effective action" in Minkowski space. Second, it becomes possible to compute various surface layer integrals as derived abstractly in [12,9,7].
There are several reasons why the asymptotics ε ց 0 of the Lagrangian and the causal action were not computed earlier. Foremost, it is unclear how to make mathematical sense of the causal action principle in the limit ε ց 0, because the constraints in the causal action principle (the so-called trace and boundedness constraints) do not have an obvious correspondence in this limit. Moreover, the fact that the causal action has a different mathematical structure than usual actions in space-time discouraged the author from taking the "naive" limit ε ց 0 of the Lagrangian seriously. It was only after the discovery of conserved surface layer integrals in [11] that the successful computation of these surface layer integrals gave a hint that the asymptotics as ε ց 0 of the Lagrangian should indeed have a direct physical significance.
The present paper is the first work in which the Lagrangian of the causal action is computed asymptotically as ε ց 0. Our results give new insight into the nature of the interaction as described by the causal action principle. One of the key findings is that there are contributions to the Lagrangian which vanish if and only if the coupled Einstein-Dirac-Maxwell equations are satisfied. Thus when minimizing the action naively ("naive" in the sense that the above-mentioned constraints in the causal action principle are disregarded), one obtains the classical field equations. The resulting contributions to the naive EL equations are more singular on the light cone than those analyzed in the continuum limit in [5,. Thus the Lagrangian has the remarkable property that it gives rise to the classical field equations in a hierarchical way several times to different degrees of the singularities of the light cone. Clearly, in an interacting system, the resulting hierarchy of equations must all be satisfied for the same values of the coupling constants. This implies that minimizing the causal action gives rise to a specific class of regularizations for which the regularization parameters satisfy all the resulting consistency conditions.
Our finding that the Lagrangian gives rise to the classical field equations several times to different degrees on the light cone is not as surprising as it might seem at first sight. Indeed, this phenomenon is closely related to the freedom in testing the EL equations. In the formalism of the continuum limit, the testing is performed by smooth variations which vanish on the diagonal (see [5, §3.5.2]). These variations have the advantage that the resulting EL equations are well-defined in the continuum limit and can therefore be analyzed in detail with mathematical rigor. But from an abstract point of view, given a minimizer of the causal action principle, the EL equations must hold for much more general variations. This becomes clear in the jet formalism introduced in [12], where the continuum limit analysis corresponds to a very special choice of the jet space J test (see [6,Section 6.2]). In particular, it should be allowed to test by varying the bosonic potentials (more precisely in direction of bosonic jets J b ; see again [6,Section 6.2]). Such variations yield EL equations with a different singular behavior on the light cone and, of course, these equations must again reduce to the classical field equations in a suitable limiting case. In general terms, on can say that the EL equations of the causal action principle give much more information on the dynamics of the system than the classical action. Consequently, when considering the limiting case of a classical interaction, the classical field equations can be derived in various ways. Taking this point of view, our findings can be regarded as a consistency check that the causal action principle makes both mathematical and physical sense.
The contributions to the Lagrangian to lower degree on the light cone can no longer be interpreted as classical field equations. Instead, they are needed in order to obtain non-trivial conserved surface layer integrals. Surface layer integrals are a generalization of surface integrals to the setting of causal fermion systems (for a general introduction see [11,Section 2.3]) As shown in [11], symmetries give rise to corresponding conservation laws for surface layer integrals. Moreover, in [12,9] other conserved surface layer integrals were discovered. Here we shall consider the symplectic form σ as found in [12,Sections 3 and 4.3] as well as the conserved surface layer inner product as first described in [9,Example 3.7]. In our setting of Minkowski space, these surface layer integrals take the form where u and v are jets describing first variations of the vacuum which preserve the EL equations (for details see [12]). The technical core of the present paper is to analyze how the electromagnetic field and the Dirac wave functions contribute to these surface layer integrals. In order to keep the calculations as simple as possible, we restrict attention to regularizations which are static and spherically symmetric. As shown in Section 4, the electromagnetic field gives the contributions (see Theorem 4.11) (where A u and A v are two electromagnetic potentials with corresponding field tensors F u and F v ). Here δ is a length scale describing the regularization (for details see [5, §1.2.1 and §4.2.5]), and c 1/2 are two real-valued constants. The integral in (1.3) is the well-known symplectic form of classical electrodynamics (see for example [1, §2.3]). Due to the absolute value in the denominator in (1.4), the surface layer inner product is semi-definite on the bosonic potentials (but it is of course degenerate on gauge orbits). The inner product (1.4) is commonly used in quantum field theory as giving rise to the scalar product on the bosonic Fock space (for example, it is used implicitly when introducing the creation and annihilation operators in [15,Sections 8.3 and 8.4]). Compared to (1.3), the surface layer inner product (1.4) contains an additional derivative and an additional factor 1/| k|. Combining these two factors gives rise to a factor plus one on the upper and minus one on the lower mass shell, thereby implementing the usual frequency splitting. It is remarkable that in the setting of causal fermion systems, the frequency splitting does not need to be put in by hand, but it follows from the theory simply by computing a surface layer integral which, by the structure of the EL equations corresponding to the causal action, is known to be conserved in time.
For the contributions by the Dirac wave functions, first variations of the vacuum are described in the jet formalism by a variation δψ of a Dirac wave function ψ, where ψ is a solution on the lower mass shell (describing a state of the Dirac sea), whereas δψ is a solution on the upper mass shell (describing a particle state). Furthermore, we restrict attention to the components of the fermionic jets which preserve the chiral symmetry (see Definition 5.16). Moreover, we only consider the situation where both ψ and δψ are solutions of the Dirac equation corresponding to the same mass m. This is motivated by the fact that we restrict attention to electromagnetic and gravitational interactions which leave the flavor unchanged. The superscripts u and v indicate which Dirac wave function is varied. Then, as shown in Section 5, the contributions to the conserved surface layer integrals are given by (see Theorem 5.13 and Proposition 5.19) with real constants c 3 and c 4 , where ω( k) := | k| 2 + m 2 is the absolute value of the corresponding frequency (and ≺.|.≻ is the usual inner product on Dirac spinors, being indefinite of signature (2,2)). The conservation of these surface layer integrals gives rise to pointwise conditions for the spinors (see (5.45) in Theorem 5.9). The inner product (1.6) is again semi-definite (see Proposition 5.21). Combining (1.4) and (1.6), we obtain a scalar product on the jet space spanned by the fermionic and bosonic jets. Our computations also reveal that the structure of the above surface layer integrals is quite different from that of the surface layer integrals describing current and energy conservation in [11,Sections 5.2 and 6.2]. Namely, while the surface layer integrals in [11] are of short range (meaning that the main contribution comes from points x and y whose distance is small on the Compton scale), the symplectic form and the surface layer inner product are essentially nonlocal and of long range in the sense that the main contributions come from points y lying on the light cone centered at x but which may be far apart from x (again on the Compton scale as measured in the reference frame distinguished by the regularization). This does not pose any problems when computing the surface layer integrals for the asymptotic incoming our outgoing states in a scattering process, and in this case the above formulas (1.3)-(1.6) again hold. But when choosing t 0 as an intermediate time while the interaction takes place, then the surface layer integral will also depend on the past and future of t 0 . We also point out that in the interacting case, the bosonic and fermionic parts of the surface layer inner product should no longer be conserved separately, because the general conservation law in [9] only applies to the whole jet space spanned by the fermionic and bosonic jets. For the symplectic form, however, there are indications that the bosonic and fermionic parts should even be conserved separately (see Proposition 4.13 and Remark 7.3). As will be worked out in detail in [13], the symplectic form and the surface layer inner product indeed give rise to a conserved scalar product on the complex Fock space of the interacting theory.
In Section 6, we consider the surface layer integral This surface layer integral is not necessarily conserved, but it is shown in [7,Section 7] that it is non-negative if one varies about a local minimizer of the causal action. Therefore, as a consistency check and in order to verify that the regularized Dirac sea configuration is indeed a minimizer, we compute this surface layer integral and find that, due to contributions by the Maxwell current, it is indeed non-negative. The paper concludes with a few remarks and an outlook (Section 7). Taken together, our results show in a surprising and compelling way that the different contributions to the Lagrangian in Minkowski space fit together consistently both with the general conservation laws of causal fermion systems and to structures present in classical field theory and quantum field theory.
We finally outline our methods. As in [5,Chapter 5] we describe the vacuum by a system of Dirac seas including leptons and quarks. Nevertheless, for simplicity we restrict attention to an interaction via electromagnetic fields and linearized gravity. Since these fields do not describe changes of flavor, in most parts of the analysis it suffices to consider a single Dirac sea of mass m. In the first step of our analysis we apply the formalism of the continuum limit (for an introduction see [5,Section 2.4]) to obtain contributions which are distributions in space-time with a δ-singularity on the light cone and which have a pole in ε. More precisely, all relevant contributions to the Lagrangian can be written as with r, s ∈ {0, 1}, where we set t = ξ 0 , r = | ξ| with ξ := y − x (here and in what follows, the symbol ≍ indicates that we restrict attention to a specific contribution). These formulas involve regularization parameters which we simply treat as effective empirical parameters. We also point out that the formalism of the continuum limit gives rise to the formulas (1.7) only away from the diagonal (i.e. for x = y). Here we simply extend these formulas in the distributional sense. Clearly, this extension is unique only up to singular contributions supported on the diagonal. It turns out that this simple method gives sensible results. For the analysis of the conserved surface layer integrals, we consider the combination given in [9,Theorem 3 (1.8) By anti-symmetrizing and symmetrizing in u and v, one gets the symplectic form (1.1) and the surface layer inner product (1.2), respectively. For the computation of this surface layer integral in Sections 4-6, we analyze integrals involving (1.7) with Fourier methods. Indeed, the distributions supported on the light cone (1.7) have a nice structure in momentum space (see for example Figure 3 on page 21). Rewriting multiplication in position space as convolution in momentum space, our task is to compute certain convolution integrals. In order to compute the fermionic surface layer integrals, we make use of the specific support properties of the Dirac wave functions and the convolution kernels in momentum space (see Figure 8 on page 51). When computing the bosonic surface layer integrals, the main difficulty is that the light-cone expansion involves unbounded line integrals (see for example Lemma 4.1). Using the causal structure of the Lagrangian, we show that these unbounded line integrals vanish. These arguments implicitly impose conditions on the admissible class of regularizations, which can be understood intuitively that the regularized objects are "supported mainly near the light cone" and "vanish approximately for spacelike distances" (see Section 4.4). Similar as worked out in [3] in the vacuum, one could analyze in detail what these conditions mean and how they can be satisfied. But this analysis goes beyond the scope of the present paper. Here we are content with showing that the highly singular contributions can be given a mathematical meaning using certain computation rules which are motivated and introduced.

Preliminaries
In this section we introduce the mathematical setup and recall a few constructions from [5].
2.1. The Fermionic Projector. We denote the points of Minkowski space by x, y ∈ M. In order to describe the vacuum, we consider the Dirac sea configuration of the standard model as introduced in [5, Chapter 5]. Thus we consider the kernel of the fermionic projector where P C is composed of the Dirac seas of the charged leptons where m β are the masses of the fermions and P vac m is the distribution The direct summand P N in (2.1) describes the neutrinos where the neutrino massesm β ≥ 0 will in general be different from the masses m β in the charged sector.
In order to describe the interaction, we first introduce the auxiliary fermionic projector by The last direct summand of P N aux has the purpose of describing the right-handed highenergy states. Next, we introduce the chiral asymmetry matrix X and the mass matrix Y by where m is an arbitrary mass parameter, and τ reg ∈ (0, 1] is a dimensionless parameter. This allows us to rewrite the vacuum fermionic projector as Now t is a solution of the Dirac equation In order to introduce the interaction, we insert an operator B into the Dirac equation, (where the tilde denotes the interaction). Finally, the fermionic projector P in the presence of the potential B is obtained by forming the sectorial projection Working in the symmetric gauge, the resulting perturbation of the Dirac operator is After including these potentials into the operator B in the Dirac equation (2.2), the distributiont is computed with the help of the causal perturbation expansion (for an introduction see [5, Section 2.1]). Before forming composite expressions in the kernel of the fermionic projector, one must introduce an ultraviolet regularization. We denote the length scale of this regularization by ε. Then the Lagrangian can be computed asymptotically as ε ց 0 in the formalism of the continuum limit (as introduced in [5, Sections 2.4 and 2.6]). We again assume the following regularization conditions (see [5, eqs (4.6.36), (4.6.38), (4.9.2) and (5.2.9), (5.2.10)]) in a weak evaluation on the light cone where the parameter p reg is in the range 0 < p reg < 2.

The Lagrangian of the Causal Action in Minkowsi Space
As in [ Using the relation (see [5, eq. (4.4

.3)])
λ xy nR± = λ xy nL∓ , it suffices to consider the summands with s = s ′ = +, In the vacuum and to leading degree five on the light cone, the eigenvalues of the close chain all have the same absolute value. Therefore, when perturbing the Lagrangian, we need to perturb each factor (|λ xy nc+ | − |λ xy n ′ c ′ + |) in (3.2), i.e.
For a convenient notation, we write the contributions to ∆L in the form where the two brackets refer to the two factors in (3.3). Many contributions to the perturbation ∆|λ xy nc+ | of the absolute values of the eigenvalues were already computed in [5]. In order to make these results applicable, it is most convenient to use the following lemma.
Lemma 3.1. The perturbation of the absolute value of the eigenvalues is related to the perturbation K nc (x, y) computed in [5,Chapter 4] by .
3.1. The Eigenvalues of the Closed Chain in the Vacuum. In the vacuum, the eigenvalues of the closed chain can be computed separately in each sector. They are given by (see [5, §4.4.2]) where L (n) Here the terms (· · · ) stand for additional contributions whose explicit form will not be needed here (for details see [2, eq. (5.3.24)]). However, it is important to keep in mind that in (3.5), these terms depend on the masses m β of the charged leptons. They coincide with the corresponding terms in (3.6) and (3.7), except that in the latter terms the masses m β are to be replaced by the neutrino massesm β . The parameter δ in (3.6) and (3.7) describes the length scale on which the regularization effects in the neutrino sector (due to the shear and the general surface states; for details see below) come into play. It scales like (for details see [5, §4.4 where p reg ∈ (0, 2) is again the power in (2.4). The length scale δ can be thought of as the Planck scale, because the gravitational coupling constant κ obtained in the continuum limit scales like (see [5,Section 4.9 and §5.4 (where ≃ means that we omit irrelevant real prefactors). A straightforward computation using (3.5)-(3.7) as well as (2.4) yields (3.8) We recall that the factor δ −2 T (0) [R,2] in (3.9) describes the effect of general surface states. This factor is essential for getting the Einstein equations in the continuum limit (see [5,  {R,0} in (3.10), on the other hand, describes the shear of the sea states. At the moment, there is no compelling reason why the regularization should involve a shear on the scale ∼ δ −2 . The shear is needed merely in order to satisfy the EL equations to the order ∼ m 2 in the vacuum. Therefore, the factor in (3.10) could be as small as We also point out that from the size of the factors δ −2 T (0) {R,0} we cannot infer how large the contributions (3.9) and (3.10) are, because there may be cancellations between the two summands inside the curly brackets.
3.2. The Contributions ∼ δ −2 · δ −2 and ∼ δ −4 · δ −4 . We now begin with the computation of different contributions to the Lagrangian, using the notation (3.4). In view of the contributions (3.9) and (3.10), the leading contribution to the Lagrangian (3.2) is of the form This contribution is in general non-zero but, as explained after (3.10), it could vanish for specific regularizations due to cancellations in the curly brackets in (3.9) and (3.10). For a better understanding of the contributions in (3.11), it is instructive to compare them to the contributions to the Lagrangian away from the light cone as computed in [8,3]. The latter contributions arise independent of the regularization and reflect the fact that, due to the different masses, the macroscopic behavior of the fermionic projector is necessarily different in the charged and neutrino sectors. The resulting contribution to the Lagrangian scales like (see [8,Section 2]  (the precise scaling depends on the size of the bilinear-dominated region as discussed in [3, Section 6]). Clearly, this contribution is much smaller than the upper bound (3.12). More precisely, the resulting contribution to the causal action per four-dimensional volume would become smaller by a scaling factor The fact that (3.11) is much larger than (3.13) suggests that when minimizing the causal action, one should try to arrange that the contribution (3.11) vanishes. The analysis in this paper gives strong indications that it is indeed physically sensible to assume that the contribution (3.11) is zero (for a detailed discussion of this point see Remark 7.1). More technically, this can be arranged as follows: First, one should keep in mind that, being a sum of squares, the expression (3.11) is non-negative. Therefore, it vanishes in a weak evaluation on the light cone only if it vanishes pointwise. Note that the contributions in (3.11) are a consequence of the the shear and general surface states (see the contributions to the eigenvalues in (3.9) and (3.10)), which are needed in order to satisfy the EL equations in the vacuum to degree four on the light cone (for details see [5, §4.4.2]). More precisely, the effect of the shear and general surface states on the absolute values of the eigenvalues is to mimic the rest masses of the charged Dirac seas. Then the contributions in (3.9) and (3.10) scale like ∼ m 2 (deg = 2), and exactly as explained in [5, §4.4.2], they can be compensated by the contributions by the rest mass in the charged sectors.
For clarity, we point out that contributions ∼ δ −2 to the closed chain are needed in order to obtain the gravitational interaction with the coupling constant κ ∼ δ 2 (see [5,Section 4.9 and §5.4.3]). However, these contributions are different from the terms in (3.11). Therefore, the assumption that (3.11) vanishes is not in conflict with a gravitational constant κ ∼ δ 2 .
Expanding to next order in 1/δ 2 , we obtain contributions where ∆|λ xy ncs | ≃ 1 δ 4 (deg = 1) . (3.14) (where ≃ again means that we omit irrelevant real prefactors). Since δ is to be chosen of the order of the Planck length, these contributions need to be taken into account. The resulting contribution to the Lagrangian scales like 3.3. The Contributions ∼ δ −4 · (J + j). We next consider the contributions where one of the brackets in (3.4) contains the contribution ∼ δ −4 given in (3.14), whereas the other bracket involves contributions by the Maxwell or Dirac currents. The perturbation by the Dirac current J is (see [5, eq. (B.2.21)]) , where in the last step we used the formula for J given in [5, beginning of Section 3.7.2]).
Here ξ := y − x, and ≺.|.≻ again denotes the spin scalar product. Likewise, the perturbation by the Maxwell current j is (see [ The resulting contributions to the Lagrangian are of the general form with a real constant c. The contribution by the Dirac current drops out of the action for the following reason: The Dirac current perturbs P (x, y) by a smooth contribution. Therefore, the resulting contribution to L(x, y) coincides (up to an irrelevant smooth prefactor) precisely with the contribution to the kernel Q(x, y) ∼ δ −4 as computed in the EL equations of the continuum limit (see [5,Proposition 1.4.3], where ψ u is the Dirac wave function giving rise to the Dirac current). Since the EL equations in the continuum limit are satisfied to the order δ −4 (see [5, §4.4.2]), the contribution ∼ δ −4 J to the Lagrangian vanishes.
The contribution by the Maxwell current behaves differently because the corresponding contribution to P (x, y) has a logarithmic pole on the light cone (see [5, §3.7.3 or §4.4.3]). This pole is removed by the microlocal chiral transformation (see [5, §4.4.4]). However, this transformation removes the logarithmic pole only in the formalism of the continuum limit where the factor ξ i in (3.16) is treated as an outer factor (for details see [5, §3.7.9 and §4.4.4]). As a consequence, the logarithmic pole remains in the regularization expansion (i.e. in the higher orders in ε/r). Hence there are contributions of the form . For the Maxwell current, this contribution drops out because it is trace-free on the sectors and vanishes in the neutrino sector.
Turning this argument around, one can take the fact that the bosonic currents must vanish in the Lagrangian to the order ∼ δ −4 · j ε/r as the reason why the bosonic currents in the standard model are all trace-free. More technically, this argument can be used as an alternative to the trace condition derived in the analysis of the field tensor terms in the ι-formalism (see [5, 3.4. The Contributions ∼ δ −4 · F . We next consider the contributions where one of the brackets in (3.4) contains the contribution ∼ δ −4 given in (3.14), whereas the other bracket contains a contribution by the Maxwell field tensor (as computed in [5, §4.6.2]). Note that these contributions vanish in the formalism of the continuum limit due to the anti-symmetry of the field tensor (because both tensor indices of the field tensor are contracted with outer factors ξ). But the contributions are in general nonzero in the ι-formalism which gives refined information on the singular behavior on the light cone (see [5, §4.2.7]). More precisely, the perturbation by the field tensor terms is (see [5,Lemma 4.6.6]) (where in the last step we integrated by parts). Hence The resulting contributions to the Lagrangian again vanish because the Maxwell field is trace-free on the sectors and vanishes in the neutrinos sector.
3.5. The Contributions ∼ F · F . We next consider the contributions where each of the brackets in (3.4) contains a contribution by the Maxwell field tensor (as computed in [5, §4.6.2]; see the formulas in Section 3.4 above).
The resulting contributions to the Lagrangian can be arranged to vanish in two different ways: One method is to impose conditions on the regularization which imply that the contributions ∼ F · F vanish in the ι-formalism. Alternatively, one can take the point of view that all the contributions obtained in the ι-formalism should be disregarded, because they depend on details of the regularization which at present are unknown and seem out of reach. However, discarding the ι-formalism makes it necessary to rely on the argument described at the end of Section 3.3 to obtain the condition that bosonic potentials must be trace-free.
It is an open question which of the above methods is physically more sensible. Fortunately, this open question does not have any consequences on the results of the present paper.
3.6. The Contributions ∼ (J + j) · (J + j). We next consider the contributions where each of the brackets in (3.4) contains a contribution by the Maxwell or Dirac current (as computed in [5,Section 3.7]; see the formulas in Section 3.3). The resulting contributions to the Lagrangian are of the general form (where, as explained in [5, §4.4.4], the logarithmic poles again vanish due to the microlocal chiral transformation). These are the Maxwell equations, where the coupling constant c is a regularization parameter which depends on the details of the regularization. We point out that in the continuum limit, the classical field equations were obtained in a completely different way (see [5,Sections 3.7,4.8 and 5.4]). The main difference is that in the continuum limit, one analyzes the EL equations of the causal action, whereas here we compute the Lagrangian and vary the classical potentials. It is remarkable that the results of both procedures give the same structural results. However, in order to get complete agreement, the coupling constant c in (3.17) must coincide with the coupling constant as computed in [5]. This poses a condition on the regularization. We remark that contributions away from the diagonal x = y can be compensated by nonlocal transformations as described in [5, Section 3.10].
3.7. The Contributions ∼ δ −4 · (F 2 + T ). We next consider the contributions where one of the brackets in (3.4) contains the contribution ∼ δ −4 given in (3.11), whereas the other bracket involves contributions by the energy-momentum tensor as computed in [5,Section 4.5]. Keeping in mind that there are also corresponding curvature terms (also computed in [5, Section 4.5]), the contribution can be written as are the energy-momentum tensors of the Dirac and Maxwell fields, respectively. This contribution vanishes if the Einstein equations hold. Exactly as explained in Section 3.6 for the Maxwell equations, here the Einstein equations appear in a quite different way than in the analysis of the continuum limit. The fact that the coupling constants must coincide poses constraints on the regularization.
There are also contributions where both brackets in (3.4) involve the energy-momentum tensor and curvature as computed in [5,Section 4.5], These contributions as well as their first variation vanish again as a consequence of the Einstein equation, provided that the regularization satisfies all consistency conditions for the coupling constants.

Computation of Bosonic Conserved Surface Layer Integrals
We now proceed with the analysis of contributions to the Lagrangian of degree three on the light cone. As we shall see, these contributions are in general non-zero. Their significance is that they give rise to physically sensible expressions for conserved surface layer integrals. More precisely, we shall compute the surface layer integral (1.8) which is composed of both the symplectic form (1.1) and the surface layer inner product (1.2). For clarity, we first consider the bosonic contributions; the fermionic contributions will be computed in Section 5 below.

Computation of Unbounded Line Integrals.
In the computation of surface layer integrals, there is the complication that the two arguments of the Lagrangian are varied differently. In particular, our task is to compute the variation for bosonic jets. Writing out the jets as variations of the wave functions, we obtain the expression − s / A P + P / A s (x, y) (4.1) (where s is the symmetric Dirac Green's operator; for basics and the notation see [5, Section 2.1] and [2, Section 2.3]). The light-cone expansion of this bi-distribution involves unbounded line integrals, as is made precise in the following lemma.
are obtained from each other by replacing the line integrals according tô Proof. In [2, Lemma F.4] a light-cone expansion involving unbounded line integrals was derived. Namely, it was shown that for any l, r ≥ 0 and any scalar potential V (here and in what follows, we assume for simplicity that the potential is a Schwartz function or a smooth function with compact support), where K is the difference of the advanced and retarded Klein-Gordon Green's distribution, Using in the last equation that the causal Green's functions are supported in the causal past respectively future, we obtain Applying this formula in (4.2) and making use of the fact that the derivatives of the step function drop out, we obtain Applying the relation ǫ(α(y 0 − x 0 )) = ǫ(α) ǫ(y 0 − x 0 ) and comparing the formulas of the light-cone expansion of the Green's functions and the fermionic projector (see [ Hence the light cone expansion of the expression (4.1) involves the line integrals Before entering the detailed computations, it might be instructive to consider the scalings, starting from the contributions to the fermionic projector as given in [5, eqs (B.5.1) and (B.5.2)]: These contributions were already taken into account in Sections 3.7 and 3.8, where they were compensated by corresponding curvature terms. With this in mind, it remains to consider the contributions obtained by a regularization expansion. More precisely, the terms with the correct scaling behavior are those of second order in ε/t, i.e.
where K 0 is the causal fundamental solution of the wave equation, One should keep in mind that in the regularization expansion, the outer factors ξ (i.e. those factors contracted with the field tensor) need not be taken into account, because they appear in the same way in the Einstein tensor, and thus they drop out exactly as explained in Sections 3.7 and 3.8 above. With this in mind, it suffices to compute the contraction of the fermionic projector with an outer factor ξ. We now compute these contributions and study their effect on the Lagrangian. For clarity, we treat the contributions which involve and do not involve logarithmic poles after each other. We point out that there are also contributions which are linear in ε/t. For clarity of presentation, we will analyze these contributions in Section 4.8 below.

Contributions With Logarithmic Poles.
We begin by computing the contributions to the fermionic projector with a logarithmic pole: where (see Figure 1)

(4.4)
Proof. Our starting point is the contribution to the fermionic projector of second order in the electromagnetic field strength as given in [5, eq. (B.5.2)], − 8iˆy Here we use the notation for the nested line integrals (for details see [5, Transforming to the new integration variables the above contribution to P (x, y) can be written as to be evaluated at The contribution symmetric in A u and A v is obtained immediately by polarizing, again to be evaluated at (4.6). Changing the sign before ∇ 2,u amounts to replacing the integrals by unbounded integrals, similar as derived in Lemma 4.1 in first order perturbation theory. In order to apply this result to second order perturbation theory, we arrange the contributions as follows, (being smooth, we can omit the so-called high-energy contributions involving three factors k or p; for details and our notation see [5, Section 2.2]). Now the terms inside the brackets involve convex line integrals, whereas the operator products outside the brackets can be handled with the help of Lemma 4.1. Using that we bring the integral corresponding to the product outside the brackets to the left and apply the replacement rules Moreover, by suitably renaming the integration variables α and β we arrange that A u is always evaluated at αy A straightforward computation gives the result.
We now compute the resulting effect on the Lagrangian: The contribution to the fermionic projector of Lemma 4.2 affects the second variation of the Lagrangian to the order ∼ δ −4 · F 2 ε 2 /t 2 by a term of the form where c is a real constant.
Proof. Computing the perturbation of the eigenvalues of the closed chain as in [5,Appendix B.5], to the considered order on the light cone we obtain for the perturbation of the Lagrangian an expression of the form with a complex constant C. Here the factor i is enforced by the fact that the left side is real-valued. Let us analyze the symmetry when exchanging x and y. Obviously, the left side is anti-symmetric. In the line integrals, exchanging x and y corresponds to the replacements From (4.4) one sees that the line integrals are anti-symmetric. This shows that the term (4.8) must be symmetric when exchanging x and y. Since K 0 is anti-symmetric, we conclude that from the factor T [0] only the anti-symmetric term ∼ iπΘ(ξ 2 ) ǫ(ξ 0 ) contributes. This gives the result.

Contributions Without Logarithmic Poles.
Proposition 4.4. The contributions to the fermionic projector involving no logarithmic poles on the light cone (i.e. all contributions except for those in Lemma 4.2) affect the second variation of the Lagrangian to the order ∼ δ −4 · F 2 ε 2 /t 2 by a term of the form where c is a real constant and (see Figure 2) Proof. Computing the perturbation of the eigenvalues of the closed chain as in [5,Appendix B.5], to the considered degree on the light cone we obtain for the perturbation of the Lagrangian an expression of the form with a real-valued function I(α, β) (to be determined below) and a real constant c.
Note that the factor i is needed in order for the expression to be real-valued. Obviously, the left side of the above equation is anti-symmetric in x and y. Moreover, the factor K 0 (ξ) is anti-symmetric. Therefore, the line integrals must be symmetric when exchanging x and y. Let us consider how this can come about. To second order in perturbation theory, we need to take into account the contribution given in [5, eq. (B.5.1)], Transforming to the integration variables α and β as in (4.5), one sees that the line integrals are symmetric in x and y (as can be verified alternatively by taking the conjugate and using that P (x, y) * = P (y, x)). Repeating the method in the proof of Lemma 4.2, after applying the replacement rules the resulting unbounded line integrals are anti-symmetric in x and y. Therefore, the second order contributions to P (x, y) do not enter (4.10). It remains to consider the contributions to P (x, y) to first order in the field strength. These are given in [5, eq. (B.2.4) and (B.2.5)], All we need here is that the integrands are linear polynomials. Therefore, the resulting line integrals in the symmetric derivatives of the Lagrangian are of the general form c t 4 iK 0 (ξ) with four parameters a, . . . , d. Since only first order perturbations of P (x, y) are involved, the sign of ∇ 2,u can be changed with the help of Lemma 4.1. Using that the resulting line integrals must be symmetric in x and y, we conclude that I(α, β) must be of the form (4.9).

Analysis in Momentum Space.
In order to gain more insight into how to compute the surface layer integrals, it is useful to transform the formulas of the previous section to momentum space. To this end, we write the contributions to the second variation of the Lagrangian as computed in Propositions 4.3 and 4.4 (as well other expressions to be introduced later) in the general form Figure 3. The kernels involving K 0 (ξ) in momentum space.
where K(ξ) is a distribution (which may involve tensor indices), and the integrand P is either I(α, β) or J(α, β) (or other similar functions to be introduced later). For the distribution K(ξ) we need to consider two essentially different cases: In Proposition 4.4, it has the form In momentum space, the distributionK 0 is supported on the mass shell. More precisely, setting p = (ω, k) and k = | p |, we havê Each factor 1/t corresponds in momentum space to i times an integration over ω. The resulting kernels for p = 1, 2 are depicted in Figure 3. In Proposition 4.3, on the other hand, the kernels are of the form Noting that δ ξ 2 is the Green's operator of the scalar wave equation, one immediately sees that its Fourier transform is ≃ PP/p 2 . Translating the factors 1/t again into ωintegrations, one obtains the kernels shown in Figure 4.
Before going on, we clarify how to handle the integration constants when rewriting factors 1/t as ω-integrals: Clearly, integrating over ω and keeping spherical symmetry gives us the freedom to add an arbitrary function of k. In position space, this corresponds to a distributional contribution supported at t = 0, which vanishes when multiplying by t. Since in position space, the distribution is supported on the light cone, this distributional contribution at t = 0 must vanish away from the origin ξ = 0. This condition can be satisfied in momentum space by demanding that the distribution be harmonic, i.e.
When integrating over ω, we always choose the integration constant such that this equation holds. In the following sections, our task is to evaluate integrals of kernels A(x, y) of the form (4.12). More precisely, the expressions of interest arê In momentum space, these expressions becomê 4.4. Conservation of Surface Layer Integrals. The conservation law for the surface layer integral (1.8) as established in [9] is based on the identitŷ Namely, it is obtained by integrating over Ω and using the anti-symmetry of the integrand (for details see [9, Proof of Theorem 3.1]). In our setting where Ω is the past of a surface {t = const} in Minkowski space, the conservation law follows already if we know that the spatial integral of (4.16) vanishes, Our strategy is to first show that, under suitable assumptions on the potentials and on the regularization, the surface layer integrals are conserved in the sense that the integral (4.17) vanishes (see Theorems 4.5 and 4.8 below). This makes it possible to express the surface layer integral in a convenient way (see Lemma 4.9 in Section 4.5). Sections 4.6 and 4.7 are then devoted to the detailed computations leading to Theorem 4.11.

Contributions Without Logarithmic Poles.
In order to explain our method and the involved assumptions, we proceed step by step, beginning with the contributions involving no logarithmic poles as computed in Proposition 4.4. Rewriting (4.17) according to (4.15) in momentum space, our task is to show that where I(α, β) is the function (4.9) andK is the Fourier transform of the kernel Recall that we consider a non-interacting region of space-time, where the potentials A u and A v describe electromagnetic waves. Therefore, choosing the Lorenz gauge, the momenta are on the mass cone. Moreover, for technical reasons we assume that the momenta are non-zero: (a) The momenta lie on the double mass cone away from the origin, As a consequence, the Lorentz inner product of the argument ofK in (4.18) simplifies The main difficulty in evaluating (4.18) is that it involves unbounded line integrals. Our first step is to show that the unbounded part of the line integrals vanishes, leaving us with an expression involving convex line integrals of the form To this end, we first consider an the integral of a polynomial in α, Carrying out the integral, we obtain a distribution supported on the hypersurface {p u ξ = 0}. Since p u is on the mass cone, this hypersurface is null. Since the distribution K(ξ) is supported on the light cone, we conclude that the hypersurface {p u ξ = 0} intersects the support of K(ξ) only on the straight line R p u . Therefore, when regularizing, by making the support of K(ξ) slightly smaller one can arrange that the supports no longer intersect. With this in mind, in what follows we shall make use of the following assumption: (b) Polynomial integrals over the whole real line (4.21) vanish in (4.18). We next consider integrals over the half line, Knowing that the integral over the whole real line vanishes, our task is to handle Heaviside functions Θ(α) or Θ(−α) in the integrand. Using (4.20), we can rewrite Handling of the spatial tensor indices.
these Heaviside functions as the characteristic function of the inner mass shell in the argument ofK, We analyze the effect of this characteristic function for different choices of the tensor indices. If i = 0 and j = 0, the distribution (4.19) simplifies to (4.13) for p = 2 as depicted on the right of Figure 3. Multiplying by the characteristic function of the inner mass shell gives a distribution which is constant inside the upper and lower mass cones and vanishes otherwise. In position space, this distribution is causal and is again supported on the light cone. If one or both tensor indices are spatial, the resulting distributionsK(p) are shown in Figure 5. Multiplying by the characteristic function of the inner mass shell gives zero. We conclude that for any choice of the tensor indices i and j, the distribution Θ(p 2 )K(p) is supported on the light cone. Therefore, we can argue just as after (4.21), leading us to the following assumption: (c) Polynomial integrals over the half line (4.22) vanish in (4.18).
Using the above assumptions (a)-(c), we can simplify (4.18) to the condition Evaluating the remaining compact line integrals gives the following result: Proof. We must consider the two cases that the momenta p u and p v lie on the same mass cone (i.e. both on the upper or both on the lower mass cone) and that they lie on different mass cones. In the first case, we see from the right of Figure 3 as well as from Figure 5 that the distributionK vanishes or is constant for all momenta −αp u − βp v . Figure 6. The functionJ(α, β).
As a consequence, the line integrals can be carried out to obtain zero, In the remaining case that p u and p v lie on different mass cones, we must make use of the fact that, due to the δ-distribution in (4.18), it suffices to consider the case p u = −p v . ThenK depends only on α − β. As a consequence, the resulting integrals vanish by symmetry,

Contributions With Logarithmic Poles.
We now turn our attention to the contributions involving logarithms as computed in Proposition 4.3. Rewriting (4.17) according to (4.15) in momentum space, our task is to show that where J(α, β) is the function in (4.4) (see Figure 1) andK is the Fourier transform of the kernel Since the distribution K(ξ) is again supported on the light cone, we can argue exactly as after (4.21) to justify the following assumption: Lemma 4.6. Using (b'), the function J in (4.23) can be replaced by the functionJ given by (see Figure 6) Proof. For clarity, we proceed in several steps. First, we subtract from J(α, β) (see Figure 1) the polynomial in α to obtain the following function: Next, we subtract the polynomial in β The resulting line integrals are still unbounded. In order to analyze the effect of the unbounded contribution of the line integrals to (4.23), it is useful to introduce the function (see Figure 7) because the function J − U has compact support. Moreover, the Fourier integral of the line integrals of U can be computed explicitly: where A is the bi-distribution Proof. Multiplying (4.26) by e ipux+ipvx , we obtain Our task is to show that this equation agrees with (4.27). To this end, we first rewrite the polynomials as derivatives, The remaining Fourier integrals can be computed with the help of the relations We thus obtain Collecting all the terms gives the result.
We now explain what this result means. The contributions containing factors δ(u) and δ(v) (and distributional derivatives of these factors) are supported on the null hypersurfaces {p u ξ = 0} and {p v ξ = 0}, respectively. Therefore, we can argue again just as for the polynomial contributions in α or β (assumption (b') above) to conclude that these contribution vanish in (4.23). The factors δ(u + v) (and distributional derivatives thereof), on the other hand, are supported on the hypersurface If p u and p v are on the same mass cone, then the momentum p u + p v is timelike, so that the hypersurface H is spacelike. As a consequence, this hypersurface intersects the support ofK(p) only in the origin. Therefore, choosing the regularization such thatK(p) vanishes at the origin, the resulting contribution to (4.23) is zero (alternatively, the vanishing of this contribution can be derived by a scaling argument when the regularization is removed). The remaining case that p u and p v are supported on different mass cones is more subtle: We first note that the space spanned by p u and p v is a two-dimensional timelike hypersurface. As a consequence, its orthogonal complement ξ ξ, p u = 0 = ξ, p v is a two-dimensional spacelike hypersurface of Minkowski space. On the left of (4.15), we integrate K(ξ) over this subspace. Similar as in Hadamard's method of descent (see for example [14, Section 5.1(b)]), the kernel obtained after carrying out this integration is again causal. With this in mind, it suffices to consider the case ξ ∈ span(p u , p v ) .
Then, due to the δ-distribution in (4.23), we only get a contribution if because p u and p v are lightlike). Hence in the limit p u → −p v we only get a contribution if ξ is tangential to the light cone. Arguing again as explained after (4.20), we are led to the following assumption: (d) Replacing the function J(α, β) in (4.23) by the function U (α, β) introduced in (4.25), the resulting line integrals vanish in (4.23). Using this assumption, we may replace the function J(α, β) in (4.23) by J − U . Hence it remains to show that Our final task is to evaluate the resulting compact line integrals. Here we proceed similar as in the proof of Theorem 4.5. However, the symmetry argument is a bit more subtle and makes it necessary to use the Maxwell equations: (e) The field tensors F u and F v satisfy the source-free Maxwell equations Proof. Due to the δ-distribution in (4.23), it again suffices to consider the case p u = − p v . We distinguish the two cases that the momenta p u and p v lie on the same mass cone and that they lie on different mass cones. In the latter case, we have p u = −p v . ThenK depends only on α − β, giving rise to the line integralŝ As is obvious from (4.29), the factor V (α, β) is odd under the replacements α ↔ β. Thus in order to prove that (4.30) vanishes, it suffices to show that the fac-torK(−(α − β) p u ) is even under the transformations α ↔ β. If both tensor indices in (4.24) are zero, this is obvious from the lower plot in Figure 4. Exactly as shown in Figure 5 for the contributions without logarithms, the spatial indices can be handled by integrating in ω and differentiating in the spatial momenta. Since this does not change the symmetry ofK about the origin, the distributionK(−(α − β) p u ) is again even under the transformations α ↔ β. This concludes the proof in the case that the momenta p u and p v lie on different mass cones.
In the remaining case that p u and p v lie on the same mass cone, the transformation α ↔ β corresponds to an inversion of the spatial component of the argument ofK. If both tensor indices in (4.24) are zero, it follows from spherical symmetry thatK is even under this transformation. Using again that V (α, β) is odd under this transformation, we obtain zero. The same argument applies if both tensor indices in (4.24) are spatial. If exactly one of the indices is spatial, integrating in ω and differentiating in the spatial momenta gives a factor k α (similar as shown in Figure 5). This factor is contracted with a field tensor. Therefore, using the Maxwell equations (e) (and similarly forF v ), we get back to the case where both tensor indices are zero. This concludes the proof.

An Integral Formula for Conserved Surface Layer Integrals.
Having proved that the surface layer integrals are time-independent, we may simplify the formula for the surface layer integrals as follows (for a related simpler computation see [11, proof of Lemma 5.5]). Lemma 4.9. Using the conservation of the surface layer integral as proved in Theorems 4.5 and 4.8, the surface layer integral can be written aŝ Proof. Differentiating the left side of (4.31) with respect to t 0 and using that the integrand is antisymmetric in the arguments x and y, we obtain precisely the expression in (4.17) evaluated at t = t 0 . Therefore, the results of Theorems 4.5 and 4.8 show that the above surface layer integral is indeed time-independent. As a consequence, denoting the spatial integrals by the surface layer integral can be written aŝ The integrals in (4.32) and (4.34) are surface layer integrals. Since A(t, t ′ ) has suitable decay properties in |t − t ′ |, these integrals are bounded uniformly in L (more precisely, these integrals exist in the Lebesgue sense provided that the electromagnetic potentials decay ∼ 1/|t|). Therefore, in the limit L → ∞ only the summand (4.33) remains, Using that A(t, t ′ ) is anti-symmetric in its arguments, we can use the same argument with the time direction reversed to obtain Taking the arithmetic mean of (4.35) and (4.36) gives the result.
4.6. Reduction to Bounded Line Integrals. Our strategy is to compute the surface layer integral using the formula of Lemma 4.9. The remaining task is to compute the expression in the last line in (4.31) for any given t, In the case i = k = 0, the Fourier transforms of these kernels are shown in Figures 3  and 4. The tensor indices can be handled again by integrating in ω and taking kderivatives (as illustrated in Figure 5). The similarity between (4.37) and (4.17) implies that the method for handling the unbounded line integrals in Section 4.4 can be applied without changes, giving the following result: For the contributions without logarithmic poles (as computed in Proposition 4.4), the kernel is given by (4.38) and Likewise, for the contributions with logarithmic poles (as computed in Proposition 4.3), the kernel is given by (4.39), and the function V (α, β) is given by (4.29).

4.7.
Computation of Bounded Line Integrals. The remaining task is to compute the bounded line integrals in (4.40). Here we face a difficulty which is in some sense complementary to the difficulties in the previous Sections 4.1-4.6, as we now explain. When analyzing the unbounded line integrals, the contributions to the line integrals for large α and/or β led to poles if y approaches x. Such difficulties for small distances can be regarded as ultraviolet problems. Accordingly, in momentum space the difficulty was to make sense of the convolution integrals for large momenta. When analyzing the bounded integrals in (4.40), however, the difficulties arise if combinations of α and β are small. More precisely, let us again assume that F u and F v have compact support. Then both arguments αy + (1 − α)x and βy + (1 − β)y must be in a fixed compact set. But if α ≈ β, we nevertheless get contributions for arbitrarily large y and x, provided that only the convex combinations above lie inside the compact set. This consideration shows that there are infrared problems if α ≈ β. Likewise, in momentum space these problems will become apparent as poles at zero momentum.

4.7.
1. An Infrared Regularization. In order to treat the infrared problems in a clean way, we introduce an infrared regularization by considering the system in finite spatial volume (as we shall see, the contribution to the surface layer integral will remain finite if the infrared regularization is removed). To this end, we insert a cutoff function into the spatial integrals. Thus let η ∈ C ∞ 0 (R 3 ) be a non-negative test function witĥ For technical simplicity, we assume that η is spherically symmetric (i.e. depends only on | x|). For a parameter R > 0 we set Then its Fourier transform isη Clearly, this is a Dirac family in the sense that Inserting η R into the spatial integrals in (4.40), we obtain the integrals where x = (0, x), and for convenience we left out the tensor indices. Using that multiplication in position space corresponds to convolution in momentum space, we get where q = (0, q). We again assume that the momenta of the field tensors lie on the mass cone (see assumption (a) on page 23). Moreover, knowing that the surface layer integrals are time independent (see Theorems 4.5 and 4.8), we only get a contribution if the frequencies of the momenta p u and p v have opposite signs. Hence, writing p u = (ω u , p u ) and p v = (ω v , p v ), it suffices to consider the cases Next, in order to clarify the R-dependence, it is useful to introduce new integration variables q, p and ∆p by q = q R and We thus obtain whereq = (0, q), and where in the last step we used (4.41) and omitted the tildes. In order to concentrate on the line integrals in this equation, we set where the momenta p u and p v are given by (4.44) and (4.43).
In order to analyze the asymptotics for large R, we first denote the arguments ofK by ω and k, Introducing the new integration variables u and v by the integration measure transforms according to We thus obtain where the boundaries of integration are given by Next, we make use of the fact thatK is homogeneous of degree minus one, i.e.
K ω, k = RK Rω, R k . For the contributions without logarithms (4.38), this is obvious by differentiating the formula in the lower plot in Figure 5 w.r.to ω. Likewise, for the contributions with logarithms, this follows from (4.39) and the explicit formula (which is obtained by integrating the formula in the lower plot in Figure 4 w.r.to ω and differentiating twice). Using (4.48) in (4.46), we obtain Now the arguments ofK can be expanded as follows, whereˆ p := p/| p | is the unit vector pointing in the direction of p.

Reduction to a Scalar Expression.
With the above transformations, we have arranged that the integrand in (4.49) converges pointwise in the limit R → ∞. Therefore, the integrals converge in this limit on every compact set. A remaining difficulty is that the integration range also depends on R (see (4.47)), making it necessary to analyze the behavior of the integrand in (4.49) for large v. SinceK is homogeneous of degree minus one and R k and Rω grow linearly in v, the integrand decays at least like v −1 log v, leading to an at most logarithmic divergence, The question is whether this divergence really occurs. This question is related to the contractions of the tensor indices, as we now explain: Being a solution of the sourcefree Maxwell equations, the field tensor has the property (e) on page 28. Moreover, from (4.43) and (4.44) we know that the momenta p u and p v coincide with the momentum (±| p|, p) up to signs and corrections of order 1/R. Since B diverges at most logarithmically, corrections of the order 1/R tend to zero as R → ∞. Therefore, we may use the computation rules 54) (where we sum over α = 1, 2, 3). Using these rules, one can get rid of all factorsp α . But factors ∆p α and q α remain. They can be treated with the following symmetry argument. By assumption, the cutoff function η is spherically symmetric. Therefore, carrying out the integrals over ∆ p and q, the spherical symmetry is broken only by the vector p. Therefore, the spatial vector and tensor indices can be decomposed aŝ with real constants a 1 , . . . , a 8 . The appearing indices p α and p β can again be treated with the help of (4.54).
In order to handle the remaining spatial tensor δ αβ , we make use of the fact that in the contributions to Lagrangian, the field tensors are contracted to outer factors ξ (see Propositions 4.3 and 4.4). Since these factors are null vectors, it is obvious that a contribution to the tensor (F u ) ij (F v ) j k which is proportional to the metric g ik drops out. In other words, only the trace-free part of the tensor comes up in our computations. Therefore, we may set After the above transformations, it remains to compute one scalar expressions. On the other hand, knowing from the above arguments that the corresponding tensor expression is spherically symmetric and only involves the trace-free part of the field tensor squared, the form of this tensor expression is determined uniquely from the scalar expression.

Completing the Computation.
For the further analysis of (4.49), it is important to take into account the symmetries under the transformations u → −u and v → −v. Therefore, it is helpful to rewrite B as . (4.56) Now a straightforward computation shows that the logarithmic terms in (4.53) drop out of the integrand. This is best verified right after applying the computation rules (4.54). Then the leading contributions as R → ∞ are of the form Thus the limit R → ∞ exists. Furthermore, due to spherical symmetry ofK andη, the value of this limit is independent of p. Moreover, a straightforward computation shows that this limit is in general non-zero. We thus obtain | p| , where our notation with −R − = R + implements that, according to (4.43), ω u and ω v have opposite signs, and s is the sign of ω u . Finally, the function g(s) is determined from the following symmetry consideration: According to (4.38), for the contributions without logarithmic poles the kernel K(ξ) is even under the transformation ξ → −ξ. Likewise, also its Fourier transformK(p) is even under sign flips p → −p (as is also obvious from the left of Figure 3). Conversely, the contributions with logarithmic poles, the kernels K andK are odd (see (4.39) and the upper plot in Figure 4). Keeping in mind that with the notation ≃ we disregard real prefactors, we thus obtain g(s) = 1 is for contributions without with logarithmic poles .
Moreover, as explained after (4.55), the correct tensor expression is obtained by the replacementF We thus obtain the following result: Proof. In the case g = 1, we obtain precisely the inner product (1.4). In the case g = is, on the other hand, the second summand in (4.57) drops out because the integrand is anti-symmetric under the replacements ω u → −ω u , ω v → −ω v and p → − p. We thus obtain where in the last step we used that the integrand is anti-symmetric. Next, using that F 0α = ∂ 0 A α − ∂ α A 0 and rewriting the derivatives as factors of the momentum variables, we obtain Next, the homogeneous Maxwell equations imply that p α (F v ) α 0 = 0 = p α (F u ) 0α . We conclude that This is precisely the symplectic form (1.3).
We point out that the prefactors in the resulting formulas may well depend on the choice of η, in agreement with our general concept that the regularization has a physical significance and may determine coupling constants, masses and prefactors in conservation laws.
4.8. The Contributions ∼ δ −4 · F 2 ε/t. We now come to the analysis of the contributions which involve one factor ε/t. Compared to the contributions in Section 4.2, these contributions are larger by a scaling factor t/ε, i.e.
We first consider the contributions with logarithmic poles.
Proposition 4.12. The contributions to the kernel of the fermionic projector as computed in Lemma 4.2 affect the second variation of the Lagrangian to the order ∼ δ −4 · F 2 ε/t by a term of the form Proof. Similar to (4.8), we can write the contribution as Compared to the proof of Proposition 4.3, the only difference is that the additional factor t/ε flips the symmetry under the replacements x ↔ y. Consequently, from the factor T (1) only the real part contributes, giving rise to the factor log |ξ 2 | ≃ log |εt|.
Proposition 4.13. Introducing an infrared regularization (4.42), the contributions to the conserved surface layer integral (1.8) of the order ∼ δ −4 · F 2 ε/t give the symplectic form,ˆt where the constant c diverges if the infrared regularization is removed.
The significance of this conservation law will be explained and discussed in Remark 7.3.
Proof fo Proposition 4.13. The proof consists of two parts: proving that the surface layer integral is conserved (4.17) and showing that the resulting integral in (4.31) gives (4.58).
In order to show conservation of the surface layer integral (4.17), we first proceed exactly as in Section 4.6 to reduce to bounded line integrals. The remaining task is to prove that the equation (4.28) holds, where K is the kernel in Proposition 4.12, As in Section 4.7, we introduce an infrared regularization and analyze the expression (4.45) with B according to (4.49). After treating the spatial tensor indices as explained after (4.54), it suffices to consider the case that both indices are timelike, i.e. Obviously, the function V (α, β) is anti-symmetric under these transformations. According to (4.56) and the formulas for Rω and R k in (4.50) and (4.51), the argument of K flips sign. According to (4.59), K and thus alsoK are even, so that K is left invariant under the above transformations. Due to spherical symmetry, the same is true for the factorsη in (4.45). It follows that A vanishes, proving that the condition for conservation (4.17) holds.
Knowing that the surface layer integral is conserved, according to Lemma 4.9 we can again compute it by evaluating the integral (4.37). Therefore, our task is to analyze again the integral in (4.31), but now with the kernel We decompose the logarithm as log |εt| = log ε + log |t| and consider the resulting terms after each other. For the contribution involving log ε, after transforming to momentum space, the resulting kernel is supported on the mass shell. Therefore, the factorK in (4.23) vanishes no matter if p u and p v are supported on the upper or lower mass shell, respectively. Hence we do not get a contribution to (4.31).
It remains to consider the contribution involving the factor log t. Transforming to momentum space, the resulting kernelK is supported also outside the mass shell. Therefore, we need to compute the integrals as explained in Section 4.7 by introducing an infrared regularization (4.42). There seems no symmetry argument showing that the resulting integrals vanish. Therefore, we get a contribution which is in general nonzero. Since V (α, β) is anti-symmetric in α and β, this contribution is anti-symmetric in u and v. This shows that we obtain again the symplectic form. However, sinceK is homogeneous of degree minus two, the contribution diverges in the limit R → ∞ when the infrared regularization is removed. This gives the result.
For the contributions without logarithms, we have the following result: Proof. Assuming that the surface layer integral is conserved, we can again compute it by evaluating the integral (4.37). Therefore, our task is to show again that (4.28) holds, but now for the kernel K(ξ) = 1 εδ 4 iK 0 (ξ) . In momentum space, this kernel is again supported on the mass shell. Therefore, we can argue exactly as after (4.60) to conclude that the resulting contributions to the surface layer integral vanish.
The remaining question is why the contributions to the surface layer integral considered in the previous proposition are conserved in time. This is a subtle and difficult question. With present knowledge, it seems necessary to arrange conservation by imposing a regularization condition. For brevity, we only sketch the construction and point out the reason for the regularization condition: Due to the additional factor t/ε, the kernel in Proposition 4.4 must be modified to Since this kernel is even in ξ, the corresponding line integrals must be odd when exchanging x with y. Therefore, instead of (4.9), the contributions to P (x, y) of first order in the field strength gives rise to the integrand involving two free parameters c 1 and c 2 . These two parameters could be determined by a lengthy computation, which we do not want to enter here. Arguing only with the symmetry properties does not determine I(α, β) up to a prefactor. As a consequence, it is impossible to evaluate the integrals obtained after the infrared regularization (4.45) and (4.56) (there seems no symmetry argument which would imply that (4.56) vanishes). Moreover, when adapting Proposition 4.4, due to the reversed symmetry of the kernel (4.61), we do need to take into account the contributions of second order in perturbation theory (4.11). This can be done similar as explained in Lemma 4.2 for the logarithmic contributions. Moreover, proceeding similar as in Lemmas 4.6 and 4.7, one could again reduce to bounded line integrals. However, it is not obvious what would be the symmetries of the resulting line integrals as well as the resulting contributions to (4.45) and (4.56). Therefore, without considerably higher computational efforts, it is impossible to compute (4.45) for the contributions without logarithms. But it can clearly be arranged by a suitable regularization condition that A vanishes.

Computation of Fermionic Conserved Surface Layer Integrals
In this section we shall compute the surface layer integral (1.8) for fermionic jets. We use the same strategy as in the computation of the bosonic surface layer integrals: We first show that, under suitable assumptions on the fermionic wave functions as well as on the regularization, the surface layer integrals are conserved, meaning that the integral (4.17) vanishes (see Section 5.4). This makes it possible to again express the surface layer integral using the formula of Lemma 4.9 (see Section 5.5). Sections 5.1-5.3 are devoted to preparatory calculations and the analysis of the scaling behavior.
5.1. The Contributions ∼ δ −4 · J 2 . In Section 3.3 we already considered the contribution where one of the brackets in (3.4) contains the contribution ∼ δ −4 given in (3.11), whereas the other bracket involves contributions by the Maxwell or Dirac currents. We now consider the contributions ∼ δ −4 which are quadratic in the currents.
We first give the scaling behavior: In order to compute the contributions in more detail, we first note that where ψ u denotes the physical wave function which is perturbed by the jet u. Exactly as explained after (3.16) for the contribution ∼ δ −4 J, the term ∇ 1,u ∇ 2,v P (x, y) does not contribute to the Lagrangian in the formalism of the continuum limit (we remark, however, that it does give rise to a finite contribution to the Lagrangian, which could be computed similar to the contribution by the Dirac current in [11,Section 5.2]). It remains to analyze the effect of (5.2) and (5.3) on the Lagrangian. Recall that one of the brackets in (3.4) involves a factor δ −4 , whereas the other bracket involves the Dirac current squared. Therefore, we need to compute the second order perturbation of the absolute values of the eigenvalues of the closed chain: Proof. We consider the different contributions after each other: (i) Second order perturbation of the eigenvalues: In the second order perturbation calculation of the eigenvalues, we consider the linear perturbation of the closed chain ∇A xy = ∇P · P * + P · ∇P * , (5.5) where for ease in notation we omitted the arguments (x, y) and wrote P * = P (y, x). Clearly, it suffices to consider the eigenvalues λ xy nL+ , because the other eigenvalues are obtained by flipping the chirality and taking the complex conjugate. The second order perturbation of λ xy nL+ is given by (see [ where F xy ncs are the spectral projectors of the closed chain of the vacuum (see [5, §2.6.1 and §5.2.4]). Note that we only consider perturbations which are diagonal in the sector index n. For ease in notation, the constant index n will be omitted in what follows. Therefore, we need to compute traces of the form Substituting (5.5) into (5.7) and expanding, we get summands involving different combinations of ∇P and ∇P * . We consider these summands after each other. The contribution involving two factors of ∇P * to the above trace is proportional to To the considered degree on the light cone, every factor ∇P * must be contracted with a factor ξ. Therefore, we may treat the terms F xy L+ / ξ and F xy c− / ξ as scalar multiples of an outer factor / ξ. Hence we may use the relation F L+ / ξ = 0 (see [5, eq. (2.6.17)]) to infer that (5.8) vanishes. The summands involving two factors of ∇P can be treated similarly.
It remains to consider the summands which involve one factor ∇P and one factor ∇P * , like for example Tr F xy L+ / ξ ∇P * F xy c− ∇P / ξ . Applying the contraction rules, one sees that either the tensor indices of the currents are contracted with each other, or else the totally anti-symmetric ǫ-tensor appears. In both cases, the resulting contributions are by a factor ε/t smaller than the contributions to be taken into account in the formalism of the continuum limit. (ii) First order perturbation of the eigenvalues: One contribution is obtained by taking the absolute value of the linearly perturbed eigenvalue, i.e. The linear perturbation of the eigenvalues is compensated by the corresponding Maxwell term (see Section 3.6). Making use of nonlocal potentials, one can even arrange that ∆λ xy cs vanishes globally (for details see [5,Section 3,10]). With this in mind, we may disregard all the terms in (5.9).
It remains to consider the contribution by a first order perturbation calculation in ∆A xy , where ∆A xy is quadratic in ∇P , i.e. ∆A xy = ∇P · ∇P * .
In this case, computing the trace in the formula ∆λ xy L+ = Tr F xy L+ ∆A xy , (5.10) either the vector components of ∇P and ∇P * are contracted to each other, or else the commutator [/ ξ, / ξ] in F xy L+ comes into play (see [5, eq. (2.6.16)]). In both cases, the resulting contributions are by a factor ε/t smaller than the contributions taken into account in the formalism of the continuum limit.
This concludes the proof.
Applying this finding to the Lagrangian immediately gives the following result: Corollary 5.2. The contributions ∼ δ −4 · J 2 to the Lagrangian vanish.
Here the factors ξ no longer need to be treated as outer factors. We now compute this contribution in detail. Since the factor δ −4 arises in the neutrino sector, whereas the currents are in the charged sectors, (3.3) simplifies to This shows in particular that we may simplify the following calculations by summing over the chiral index c. − Tr F xy n+ (∇P ) P * F xy n− P (∇P * ) (5.12) − Tr F xy n+ P (∇P * ) F xy n− (∇P ) P * (5.13) + Tr I n P * P (∇P ) (∇P * ) , (5.14) valid for any n ∈ {2, . . . , 8}.
Proof. We again leave out the sector index n. Also, for ease in notation, we again omit the arguments (x, y) and write P * = P (y, x). We must be careful to apply only those computation rules which are valid to higher order in ε/t. In particular, we may still use the relations F xy L+ P = P F xy R− , F xy L+ P * = P * F xy R− . Moreover, the products F xy L− P and F xy L− P * are multiples of each other, F xy L− P = c F xy L− P * with c ∈ C . (5.15) This follows from the general fact that the operator products on the left and right have the same one-dimensional image. Moreover, they are both vectorial. Hence this vector must be the same null vector. Alternatively, the relation (5.15) can be verified by a direct computation. Indeed, (5.16) showing that (5.15) holds with For clarity, we remark that in the formalism of the continuum limit, these formulas simplify to We again use the formula (5.6) for the second order perturbation of the eigenvalue λ xy nL+ to obtain ∆ λ xy Since we saw after (5.7) that the trace vanishes in the continuum limit, it gives rise to a factor ε/t. Therefore, the other factors can be computed in the formalism of the continuum limit. Thus in ( * ) we could use the explicit form of the eigenvalues in the continuum limit (see [5, §3.6.1]). Adding the similar formula for the righthanded eigenvalue, we obtain Collecting all the contributions gives the result.
For the computation of the resulting contributions to the Lagrangian, we shall make essential use of the following symmetry argument: Lemma 5.4. Let S be an expression linear in P and P * which may contain factors of ξ. Assume that the expression is even under the replacements P ↔ P * . Then the combination of derivatives is odd (even) under the replacements x ↔ y if the number of factors ξ is even (respectively odd).
Proof. Since P (x, y) * = P (y, x), the expression S is even (odd) under the replacements x ↔ y if the number of factor ξ is odd (even). Clearly, the combination of derivatives in (5.25) reverses the symmetry from odd to even and vice versa. This gives the result.
Proposition 5.5. The contributions ∼ δ −4 · J 2 ε/t to the Lagrangian have the form where the J lk (x, y) has the components whereL = R andR = L (moreover, the indices 1 and 2 again refer to the derivatives with respect to x and y, respectively).
In ( * ), the fact that the operator F xy L+ has a one-dimensional image made it possible to factorize the trace (for details on this method see [2,Appendix G.2]). Since the factor (P * ) 2 already gives the scaling factor ε/t, the resulting traces can be computed in the formalism of the continuum limit. This gives (see (5.17) and (5.21) or [5, eq. (2.6.16)]) Tr χ L ∇P Tr χ R ∇P P * 2 (5.34) Tr (∇P L ) (∇P L ) P * 2 (5.36) Finally, we add the resulting formula for the right-handed component. It remains to consider the summands (5.12)-(5.14) involving one factor ∇P and one factor ∇P * . Rewriting (5.12) as (5.12) = − Tr F xy + (∇P ) F xy + P * P (∇P * ) = − Tr F xy + (∇P ) F xy + (∇P * ) Tr F xy L+ P * P , one sees that (5.12) is symmetric under the replacement ∇P ↔ ∇P * . As a consequence, the symmetry argument of Lemma 5.4 applies. The counting of the number of factors ξ is already taken into account by our expansion in powers of ε/t. Representing the contribution in the form (5.26), the expression J kl (x, y) is odd under the replacements x ↔ y. Since the factor K 0 (ξ) in (5.26) is also odd, we conclude that the contribution on the right of (5.26) is necessarily even. But the left side of (5.26) is odd. Therefore, these contributions must vanish in (5.26).
The term (5.13) can be treated similarly after rewriting it according to It remains to consider the summand (5.14). Since contributions symmetric under the replacement ∇P ↔ ∇P * again vanish, it suffices to consider the contribution anti-symmetrized under this replacement, 1 4 Tr P * P ∇P, ∇P * where in the last step we used that P and P * are vectorial, so that their anticommutator commutes with ∇P and ∇P * . Finally, we specialize the above formulas for a spherically symmetric regularization. Then Collecting all the results and expressing ∇P and its adjoint according to ( Lemma 5.6. For any test function h ∈ C ∞ 0 (R 4 ) (or in our applications a spinorial test function h ∈ C ∞ 0 (R 4 , C 4 )) and for any momentum q with q 2 > 0 and any mass parameter m > 0, Proof. Due to Lorentz symmetry we may assume that q = (Ω, 0) with Ω = 0. Moreover, we first consider the situation that h(p − q) is a constant χ. Then, using spherical symmetry of the resulting integrals, This gives the result if h is a constant. Moreover, the above computation shows that if q 2 ≈ m 2 , one only gets a contribution for p ≈ 0 (as one also sees graphically by varying q on the right in Figure 8). Therefore, we may expand h(p − q) in a Taylor polynomial around p = 0 to obtain the result.
Lemma 5.7. For any test function h and any momentum q inside the upper mass cone {q 2 > 0 and q 0 > 0}, Moreover, the curly brackets in (5.41) vanish quadratically on the mass cone, i.e.
Proof. In the case q 2 ≥ m 2 , in (5.41) one integrates only over p inside the upper mass cone. Likewise, if q 2 < m 2 one integrates over p inside the lower mass cone. We consider these two cases after each other and begin with the case q 2 ≥ m 2 . If in (5.40) one replaces the factorK 0 by a δ-distribution supported on the boundary of the upper mass cone, one gets for any r ∈ R 4 , Next, we rewrite the Heaviside function in the integrand in (5.41) as an integral over a δ-distribution. Namely, for p = (ω, p) with ω > 0, where in the last step we set l = (ℓ, 0) ∈ R 4 . Substituting this relation into the integral in (5.41) and using (5.42), we obtain Computing the ℓ-integral gives the result in the case q 2 ≥ m 2 .
In the remaining case q 2 < m 2 , we proceed similarly for the Heaviside function and the δ-distribution in the lower mass cone, valid for all r inside the upper mass cone, Moreover, for p = (ω, p) with ω < 0, Again carrying out the ℓ-integral completes the proof.
In the next lemma, we show that inserting a factor ω in the integrand in (5.41) makes the contribution smaller by one order on the mass cone: Lemma 5.8. For any momentum q inside the upper mass cone {q 2 > 0 and q 0 > 0}, Proof. We proceed as in the proof of Lemma 5.7 and estimate the resulting contributions. First, inserting a factor ω in (5.42), a calculation similar to (5.40) shows In the case q 2 ≥ m 2 , a computation similar to (5.43) yieldŝ The case q 2 < m 2 can be treated similarly. 5.4. Conservation of Surface Layer Integrals. According to Proposition 5.5, the integrand of the surface layer integral is of the form where J kl (x, y) is real-valued and symmetric, i.e.
As we shall see, the surface layer integrals are conserved only under certain assumptions on the fermionic jets. In the applications, these assumptions pose conditions on the nature of microscopic mixing. This is the main result of this section: Theorem 5.9. Assume that all fermionic jets satisfy for all x, y ∈ M and α = 1, 2, 3 the conditions (where in each equation we can choose the plus or the minus sign independently). Then the surface layer integrals are conserved.
The significance of the condition (5.45) will be explained and discussed in detail in Section 5.4.3 below. We now enter the proof of this theorem, which will be completed at the end of Section 5.4.3. It is useful to write the y-dependence of J kl (x, y) symbolically as ψ(y) · φ(y) , (5.46) where ψ and φ are two solutions of the Dirac equation. Since the spinorial character of the wave functions is of no relevance for what follows, it is preferable to consider ψ and φ as solutions of the Klein-Gordon equation of mass m > 0. Thus in momentum space, bothψ andφ are supported on the mass shell. Rewriting the integral (4.16) in momentum space, our task is to analyze the integral For the following analysis, it is convenient to decomposeK into the so-called equal time and zero momentum contributions, Lemma 5.10. The contribution by K et to the surface layer integrals is given bŷ Proof. Transforming (5.48) back to position space, one finds Since the conservation laws follow from the causal action principle (see [12] and [9]), one can say that (5.56) gives information on the interaction described by the causal action principle. Of course, it is an important consistency check that the condition (5.56) can be satisfied in physically relevant situations. Indeed, the condition (5.56) can be arranged in various ways, as we now discuss. First, one should keep in mind that we get a contribution only if both Dirac solutions ψ, φ describing the y-dependence of J kl (x, y) in (5.46) are Dirac solutions which are supported both on the upper mass shell or both on the lower mass shell. Consequently, we only need to take into account the contributions in (5.31) and (5.32) where the two wave functions at y (and similarly at x) have frequencies of opposite signs. Moreover, due to momentum conservation, we only get a contribution if the spatial momenta add up to zero. For example, in the contribution the corresponding momenta must satisfy the relation Time independence of this contribution means that also the frequencies must cancel each other, i.e.
If the left and right side of (5.57) vanish separately, then (5.58) is also satisfied (exactly as explained in the proof of Lemma 5.10 after (5.53)). However, if p δψ u − p ψ u = 0, then (5.58) will in general be violated. This is the reason why (5.56) imposes conditions on the fermionic jets. One way of satisfying (5.56) is to ensure the implication For example, one can assume that there is a function Ω : This leaves a lot of freedom to choose the fermionic jets (δψ u , ψ u ). More specifically, one could consider the situation that the momenta p δψ u and p ψ u are related to each other by a linear transformation A : R 3 → R 3 , But clearly, there are many other ways of choosing the jets such that the implication (5.59) holds. Another strategy for satisfying (5.56) is to arrange that the contributions in (5.31) and (5.32) vanish or cancel each other. This has the advantage that J αβ (x, y) can be arranged to vanish pointwise, making it unnecessary to argue with momentum conservation. In particular, we not only satisfy the conservation law for the surface layer integral (4.17), but even the pointwise relation (4.16) which should hold in order to comply with the EL equations. In Theorem 5.9, such pointwise conditions are implemented, as is shown in the following lemma: Tr χ c σ 0α ∇P Tr χc σ 0β ∇P ξ α ξ β The condition J αβ = 0 gives rise to quadratic equations. Since all the above traces are in general complex (note that ∇P may involve arbitrary phase factors), the only sensible method for satisfying these quadratic equations by linear relations seems to impose that pairs of these traces are multiples of each other, Tr σ 0α ∇P = ± Tr γ α ∇P and Tr γ 5 σ 0α ∇P = ±i Tr γ 5 γ α ∇P .
This lemma also completes the proof of Theorem 5.9. We finally remark that if the fragmentation of the universal measure is taken into account (see [12,Section 5] or the mechanism of microscopic mixing in [4]), then (5.56) must be satisfied only after summing over the subsystems. This gives many more possibilities to satisfy the conditions for conservation. For example, one could arrange that, after summing over the subsystems, the tensor J αβ is spherically symmetric, implying that its contractions in (5.56) vanish.

5.5.
Computation of the Conserved Surface Layer Integral. Knowing that the surface layer integral is conserved, we may compute it using again the formula of Lemma 4.9. This gives the following result: Theorem 5.13. Under the assumptions of Theorem 5.9, for the contribution (5.26) the surface layer integral is computed bŷ For the proof, we need to compute the expression (4.37). The factor (y 0 − t) gives rise to a factor t in (5.44), so that we need to consider the kernel We again work in momentum space and decomposeK similar to (5.48) and (5.49) into the equal time and zero momentum contributions, We again analyze these contributions after each other.

The Equal Time Contribution.
Lemma 5.14. Under the assumptions of Theorem 5.9, the equal time contribution to the surface layer integrals vanishes.
Proof. Transforming (5.65) back to position space, one finds Under the assumptions (5.45), the terms (5.31) and (5.32) cancel each other, so that J αβ is zero. This gives the result.

The Zero Momentum Contribution.
Lemma 5.15. The contribution by the kernel K zm to the conserved surface layer integrals is given by (5.60) with G s,s ′ and H s,s ′ according to in (5.61)-(5.64).
Proof. We again write the y-dependence of J kl according to (5.46) as the product of two Dirac solutions. Moreover, we work again in momentum space (5.54). Exactly as explained after (5.47), it again suffices to consider the case that momenta q and q − p lie both on the upper or both on the lower mass shell. Using (5.66) and integrating by parts, we obtain for the inner integral in (5.54) where in the last step we again used the notation (5.55). Using the assumption (5.45), the spatial components J αβ vanish. Therefore, at least one tensor index is zero. Setting for example the index k to zero, the corresponding derivative can be carried out and integrated by parts, (in the last step we used that on the mass cone, p 0 /k = ǫ(p 0 )). Computing the last integral with the help of Lemma 5.6, we obtain (again in the case that the tensor index k = 0)ˆd In order to relate h(−q) back to the Dirac wave function φ, we evaluate (5.55) at p = 0, multiply by e −iqx and integrate over the frequency. We thus obtain where ω( q) := | q| 2 + m 2 . Combining this equation with (5.54) and (5.67), we obtain where in ( * ) we made use of the assumption that the frequencies ofψ andφ have opposite signs. Collecting all the contributions gives the result.
This concludes the proof of Theorem 5.13 5.5.3. Building in a Chiral Symmetry. Since the result of Theorem 5.13 is rather complicated, it is helpful to simplify the setting by imposing additional assumptions on the form of the fermionic jets. Since we restrict attention to Maxwell fields which are vectorial, we should consequently only consider the perturbations by the Dirac wave functions which preserve the chiral degeneracy of the eigenvalues of the closed chain.
Under this assumption, we can simplify the result of Theorem 5.13 as follows.
Proposition 5.17. If the fermionic jets preserve the chirality, then the formula for the function H s,s ′ given in (5.63) and (5.64) can be written alternatively as (the sign ± in (5.71) and (5.72) is to be chosen in agreement with the left equation in (5.45)).
The main point of this proposition is that the matrices σ 0α in (5.63) and (5.64) have been replaced by Dirac matrices γ α . This will be crucial for the analysis of the positivity properties of the surface layer inner product in Section 5.6. In preparation for the proof, we observe that the term (5.64) vanishes as a consequence of the Dirac equation: On the other hand, integration by parts giveŝ Taking the mean of (5.73) and (5.74) giveŝ Due to momentum conservation, we only get a contribution if the momenta ofφ andψ coincide. Thus the corresponding frequencies coincide up to signs. The combination in (5.75) vanishes if the frequencies have the same sign. This gives the result.
Proof of Proposition 5.17. It remains to simplify the term (5.63). We first note that this term can be written aŝ Tr σ 0α P P * d 4 y .
Thus we may replace the factor σ 0α in (5.77) by ∓γ α , This gives the formula for H s,s ′ in (5.71) and (5.72), concluding the proof. 5.6. Exciting the Dirac Sea and Positivity of (.|.). We now consider the special case that ψ u and ψ v have negative frequencies, whereas δψ u and δψ v have positive frequencies. Then the result of Theorem 5.13 simplifies tô Due to the integration over y, the momenta of ψ u and δψ v in (5.79) coincide. Hence their frequencies coincide except for a sign, implying that the derivatives ∂ 2 0 can act just as well on the function δψ v . Arguing similarly for the other summands, one sees that G is anti-symmetric in u and v, whereas H is symmetric. Therefore, G and H give rise to the symplectic form and the surface layer inner product, respectively. Rewriting the spatial integrals in momentum space gives the following result.
Proposition 5.19. Assume that ψ u and ψ v are Dirac solutions on the lower mass shell, whereas δψ u and δψ v are solutions on the upper mass shell. Then the symplectic form σ(., .) and the surface layer inner product (.|.) are given by where ω( k) := | k| 2 + m 2 (and all functions are evaluated at t = 0).
Proof. The formula for the symplectic form (5.87) follows immediately by rewriting the spatial integrals in (5.78) as integrals over momentum space and using (5.79)-(5.82). The formula for the inner product (5.88) follows similarly if one removes the Dirac matrices inside the spinorial expectation values using Lemma 5.20 below. This concludes the proof.
Proof. We begin with the expression involving a derivative on the right hand side, The last line vanishes ifφ andψ have the same frequency, giving the result. Proof. We evaluate (5.88) for u = v. Since ψ and δψ are supported on the lower and upper mass shell, respectively, the real part in (5.88) is necessarily negative. In order to show that the curly brackets in (5.88) are non-negative, we consider the worst possible case that k· q = −| k| | q|. In this case, the curly brackets in (5.88) simplify to | q| 2 ω( q) + | k| 2 ω( k) − | k| | q| ω( q) + ω( k) = | q| − | k| | q| ω( q) − | k| ω( k) .
Since the function ω( k) is strictly monotone increasing in | k|, the two square brackets always have the same sign, and they vanish if and only if | q| = | k|. This concludes the proof.
This result shows that, by a suitable choice of the sign in the first equation in (5.45), one can arrange that the surface layer inner product is positive definite.

5.7.
The Contributions ∼ δ −4 · jJ and ∼ δ −4 · j 2 . We remark that there are also contributions ∼ δ −4 · jJ which are linear in the Dirac and linear in the Maxwell current, as well as contributions ∼ δ −4 · j 2 quadratic in the Maxwell current (as well as expansions of these contributions in powers of ε/t). Moreover, there are contributions involving j or higher derivatives of the field tensor. All these contributions have a different structure than the contributions quadratic in the Dirac current (mainly because the Maxwell current gives rise to unbounded line integrals as explained in Section 4.1 for the field tensor terms). As a consequence, it seems impossible that these contributions partially cancel contributions ∼ J 2 . With this in mind, for brevity we shall not enter the analysis of these contributions.

Computation of a Positive Surface Layer Integral
In [7] positive functionals in space-time were derived. We shall now verify that in Minkowski space, these functionals are indeed positive. There are the two positive functionals involving volume integrals (see [ Since in the formalism of the continuum limit, the functions ℓ(x) and L(x, y) vanish in the Minkowski vacuum, the corresponding measure ρ clearly is a minimizer. Therefore, the inequalities (6.1) and (6.2) obviously hold. The positivity of the surface layer integral (6.3) is less obvious. Therefore, it will be instructive to compute this surface layer integral in Minkowski space and to verify that it is indeed positive. We begin with the contributions by the Maxwell current (as we shall see below, these are indeed the dominant contributions). The corresponding contribution to the fermionic projector is given by (see [5, eq. (D.0.7)]) 1 4 Tr (/ ξ ∆P (x, y)) ≍ −2ˆy [0] (x, y) . (6.4) This contribution clearly dominates the contributions involving the Dirac current, because the Dirac current does not lead to unbounded line integrals, giving rise to contributions to L(x, y) which are less singular at y = x. We conclude that the surface layer integral in (6.3) is indeed positive. We close with two remarks. We first point out that, since the currents have contributions for space-like momenta (see Figure 8), we cannot again use the arguments in Sections 4.4 and 4.6 to conclude that the unbounded line integrals drop out of the surface layer integrals. It seems impossible to avoid the pole of order t −10 in (6.6). We also remark that the contribution (6.6) does not seem to have any direct physical significance. But it could nevertheless be useful for analytic studies and estimates of the causal action principle.

Remarks
We close with a few remarks. The first two remarks point to possible modifications of our constructions which might be worth exploring in detail in the future.
Remark 7.1. (Vanishing of Contributions |λ xy ncs | ∼ δ −2 ) In Section 3.2 it was explained why it is natural and desirable to assume that in the vacuum, the absolute values of the closed chain agree up to contributions of the order δ −4 (see (3.14)). But we would like to point out that the argument leading to this assumption was not compelling. We now outline how our constructions and results would have to be modified if contributions ∆|λ ncs | ∼ δ −2 (deg = 2) (7.1) were present. The fact that this would no longer lead to physically sensible results can serve as a further explanation why the contributions (7.1) must indeed be zero. First, instead of the contributions ∼ δ −4 · F 2 ε/t in Section 4.8, one would have contributions ∼ δ −2 · F 2 ε/t. These contributions differ from the contributions ∼ δ −4 · F 2 ε 2 /t 2 in Section 4.2 only by a constant prefactor δ 2 /ε 2 . Therefore, the results of Theorem 4.11 would remain valid except for this prefactor, giving rise to the scalings σ(u, v), u|v ∼ 1 ε 2 δ 2 . (7. 2) The advantage of this procedure is that it becomes unnecessary to introduce a regularization condition to arrange the conservation of the surface layer integral ∼ δ −4 ·F 2 ε/t (see the discussion after Proposition 4.14). The drawback of having contributions ∆|λ ncs | ∼ δ −2 is that the fermionic surface layer integrals would no longer have the correct scaling. In particular, the contributions in Section 5.2 would be of the order ∼ δ −2 · J 2 ε/t, being by a scaling factor ε/t smaller than the bosonic surface layer integrals in (7.2). A further regularization expansion or an expansion in powers of δ −2 would not be helpful at this point, because this would make the resulting contributions to the fermionic surface layer integrals even smaller. We conclude that without assuming that the contributions (7.1) vanish, the bosonic and fermionic components of the surface layer integral would necessarily have a different scaling behavior in δ/ε. This seems to be an obstruction for getting a scalar product on the jets having bosonic and fermionic components as needed for getting the connection to quantum field theory [13]. ♦

Remark 7.2. (Shear and General Surface States only in Neutrino Sector)
Following the procedure in [5], we here consider the shear and general surface states only in the neutrino sector. This procedure seems natural and has the advantage that it also breaks the chiral symmetry in the neutrino sector, as is needed in order to explain why the neutrinos do not take part in the strong and electromagnetic interactions. However, one could introduce shear and general surface states also in the charged sectors, provided that they preserve the chiral symmetry. For the results in [5], this procedure would not have any influence on the results because in the analysis of the continuum limit, only the difference of the regularization effects in the charged sectors and the neutrino sector comes into play. This is the main reason why in [5] we could simply disregard shear and general surface states in the charged sectors. For the computations in this paper, having shear and general surface states in the charged sectors would make a substantial difference. Namely, when expanding in powers of δ −2 , these factors could also appear in the eigenvalues of the charged sectors. For example, in addition to the contributions to the Lagrangian ∼ δ −4 · J 2 , there would also be contributions ∼ δ −4 J · J (where the dot again refers to the notation (3.4)). These additional contributions would not change the general structure of our computations, but they would affect the quantitative details in such a way that it is difficult to predict how our results would have to be modified. ♦ We finally mention a question which our analysis did not answer: Remark 7.3. (Separate Conservation of the Bosonic Symplectic Form) In Section 4.8 a conservation law for the bosonic symplectic form was derived (see Proposition 4.13). We now discuss the significance of this conservation law. The fact that the bosonic symplectic form appears several times to different degrees on the light cone (cf. Proposition 4.13 and Theorem 4.11) can be understood similar as explained for the hierarchy of classical field equations in the introduction. The expression in (4.58) has the surprising feature that it diverges if the infrared regularization is removed. This does not pose any principal problem, because one can consider the system in finite spatial volume and take the infinite volume limit. But the infrared divergence in (4.58) implies that the bosonic and fermionic parts of the symplectic form must be conserved independently. While this result seems physically sensible, we would like to point out that this result is not compelling. Indeed, it is conceivable that for physical regularizations, the constant c in (4.58) is zero. The main unknown at the present stage is why the contributions of the order δ −4 · F 2 ε/t are conserved (as discussed after Proposition 4.14). In order to clarify the situation, one would have to compute the effect of the terms without logarithms in more detail and analyze the question if one really needs a regularization condition in order to ensure the conservation of the surface layer integral. If the answer is affirmative, one would have to analyze which condition on the regularization is required. This would lead to the question if this regularization condition also implies that the contribution (4.58) vanishes. ♦ Generally speaking, it would be desirable to have more detailed information on the regularization of the physical vacuum. The main open question seems to be how the regularized neutrino sector looks like (see Remarks 7.1 and 7.2). For example, our results might change considerably if the neutrino sector contained additional Dirac seas corresponding to yet unobserved particles. Clearly, the uncertainties in the neutrino sector are related to the fact that particles in the neutrino sector interact only very weakly. Hopefully, future experimental input will clarify the situation.